updates

content/Math416/Math416_L18.md

@@ -26,21 +26,27 @@ $$
 \int_{C(z_0,r)} f(z) dz = \sum_{n=-\infty}^{\infty} c_n \int_{C(z_0,r)} (z-z_0)^n dz
 $$
 
-> $$
-\int_{C(z_0,r)} (z-z_0)^n dz = \begin{cases}
-    2\pi i, & n=-1 \\
-    0, & n\neq -1
-\end{cases}$$
-> Proof:
-> $\gamma(t)=z_0+re^{it}, t\in[0,2\pi]$
-> $$\begin{aligned}
+<details>
+<summary>Additional Proof</summary>
+
+$$
+\int_{C(z_0,r)} (z-z_0)^n dz = \begin{cases} 2\pi i, & n=-1 \\0, & n\neq -1\end{cases}
+$$
+
+Proof:
+
+$\gamma(t)=z_0+re^{it}, t\in[0,2\pi]$
+$$
+\begin{aligned}
 \int_{C(z_0,r)} (z-z_0)^n dz &= \int_0^{2\pi} (z_0+re^{it}-z_0)^n ire^{it} dt \\
 &= ir^{n+1} \int_0^{2\pi} e^{i(n+1)t} dt \\
 &= \begin{cases}
     2\pi i, & n=-1 \\
     \int_0^{2\pi} e^{i(n+1)t} dt = \frac{1}{i(n+1)}e^{i(n+1)t}\Big|_0^{2\pi} = 0, & n\neq -1
 \end{cases}
-\end{aligned}$$
+\end{aligned}
+$$
+</details>
 
 So,